Propagation fronts in a simplified model of tumor growth with degenerate cross-dependent self-diffusivity
Motivated by tumor growth in Cancer Biology, we provide a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby–Gawlinski model∂tU=Uf(U)−dV,∂tV=∂xf(U)∂xV+rVf(V),where f(u)=1−u and the parameters d,r are positive. Denoting by (U,V) the traveling wave profile and...
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| Published in | Nonlinear analysis: real world applications Vol. 63; p. 103387 |
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| Format | Journal Article |
| Language | English |
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01.02.2022
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| Abstract | Motivated by tumor growth in Cancer Biology, we provide a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby–Gawlinski model∂tU=Uf(U)−dV,∂tV=∂xf(U)∂xV+rVf(V),where f(u)=1−u and the parameters d,r are positive. Denoting by (U,V) the traveling wave profile and by (U±,V±) its asymptotic states at ±∞, we investigate existence in the regimes d>1:(U−,V−)=(0,1)and(U+,V+)=(1,0),d<1:(U−,V−)=(1−d,1)and(U+,V+)=(1,0),which are called, respectively, homogeneous invasion and heterogeneous invasion. In both cases, we prove that a propagating front exists whenever the speed parameter c is strictly positive. We also derive an accurate approximation of the front profile in the singular limit c→0. |
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| AbstractList | Motivated by tumor growth in Cancer Biology, we provide a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby-Gawlinski modelwhere f (u) = 1 -u and the parameters d, r are positive. Denoting by (U, V) the traveling wave profile and by (U±, V±) its asymptotic states at ±∞, we investigate existence in the regimesand U+, V+ = 1, 0 , which are called, respectively, homogeneous invasion and heterogeneous invasion. In both cases, we prove that a propagating front exists whenever the speed parameter c is strictly positive. We also derive an accurate approximation of the front profile in the singular limit c → 0. Motivated by tumor growth in Cancer Biology, we provide a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby–Gawlinski model∂tU=Uf(U)−dV,∂tV=∂xf(U)∂xV+rVf(V),where f(u)=1−u and the parameters d,r are positive. Denoting by (U,V) the traveling wave profile and by (U±,V±) its asymptotic states at ±∞, we investigate existence in the regimes d>1:(U−,V−)=(0,1)and(U+,V+)=(1,0),d<1:(U−,V−)=(1−d,1)and(U+,V+)=(1,0),which are called, respectively, homogeneous invasion and heterogeneous invasion. In both cases, we prove that a propagating front exists whenever the speed parameter c is strictly positive. We also derive an accurate approximation of the front profile in the singular limit c→0. |
| ArticleNumber | 103387 |
| Author | Mascia, Corrado Gallay, Thierry |
| Author_xml | – sequence: 1 givenname: Thierry surname: Gallay fullname: Gallay, Thierry email: Thierry.Gallay@univ-grenoble-alpes.fr organization: Institut Fourier, Université Grenoble Alpes, CNRS, 100 rue des Maths 38610 Gières, France – sequence: 2 givenname: Corrado orcidid: 0000-0002-7294-492X surname: Mascia fullname: Mascia, Corrado email: corrado.mascia@uniroma1.it organization: Dipartimento di Matematica Guido Castelnuovo, Sapienza, Università di Roma, Piazzale Aldo Moro 5 00185 Roma, Italy |
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| Keywords | Degenerate diffusion Reaction–diffusion systems Cross-dependent self-diffusivity Traveling wave solutions Singular perturbation 2010 Mathematics Subject Classification. 35C07 35K57 34D10 92-10 Reaction-diffusion systems cross-dependent self-diffusivity traveling wave solutions degenerate diffusion singular perturbation traveling wave solutions singular perturbation cross-dependent self-diffusivity degenerate diffusion 2010 Mathematics Subject Classification. 35C07 35K57 34D10 92-10 Reaction-diffusion systems |
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| Snippet | Motivated by tumor growth in Cancer Biology, we provide a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby–Gawlinski... Motivated by tumor growth in Cancer Biology, we provide a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby-Gawlinski... |
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| SubjectTerms | Cross-dependent self-diffusivity Degenerate diffusion Mathematics Reaction–diffusion systems Singular perturbation Traveling wave solutions |
| Title | Propagation fronts in a simplified model of tumor growth with degenerate cross-dependent self-diffusivity |
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